Abstract
In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier–Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier–Stokes equations have no Type I blowups.
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References
Caffarelli, L., Kohn, R.-V., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. XXXV, 771–831 (1982)
Chae, D., Lee, J.: On the regularity of the axisymmetric solutions of the Navier–Stokes equations. Math. Z. 239, 645–671 (2002)
Chen, C., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier–Stokes equations. Discret. Contin. Dyn. Syst. - A 37(4), 1923–1939 (2017)
Chen, C., Strain, R.M., Yau, H., Tsai, T.: Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations II. Commun. Part. Diff. Equa. 34, 203–232 (2009)
Escauriaza, L., Seregin, G., Sverak, V.: \(L_{3,\infty }\)-solutions to the Navier–Stokes equations and backward uniqueness, Uspekhi Matematicheskih Nauk, v. 58, 2(350), pp. 3–44. English translation in Russian Mathematical Surveys, 58 2, pp. 211–250 (2003)
Kang, K.: Regularity of axially symmetric flows in half-space in three dimensions. SIAM J. Math. Anal. 35(6), 1636–1643 (2004)
Ladyzhenskaya, O.A.: On unique solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations under the axial symmetry. Zap. Nauchn. Sem. LOMI 7, 155–177 (1968)
Lei, Z., Zhang, Q.: A Liouville theorem for the axially symmetric Navier–Stokes equations. J. Funct. Anal. 261, 2323–2345 (2011)
Leonardi, S., Malek, J., Necas, J., Pokorny, M.: On axially simmetric flows in \(\mathbb{R}^3\). ZAA 18, 639–649 (1999)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Lin, F.-H.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)
Nazarov, A.I., Uraltseva, N.N.: The Harnack inequality and related properties for solutions to elliptic and parabolic equations with divergence-free lower-order coefficients. St. Petersb. Math. J. 23(1), 93–115 (2012)
Neustupa, J., Pokorny, M.: Axisymmetric flow of Navier–Stokes fluid in the whole space with non-zero angular velocity compnents. Math. Bohem. 126, 469–481 (2001)
Scheffer, V.: Partial regularity of solutions to the Navier–Stokes equations. Pac. J. Math. 66, 535–552 (1976)
Scheffer, V.: Hausdorff measure and the Navier–Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)
Seregin, G.: Estimates of suitable weak solutions to the Navier–Stokes equations in critical Morrey spaces. Zapiski Nauchn. Seminar, POMI 336, 199–210 (2006)
Seregin, G.: Note on bounded scale-invariant quantities for the Navier–Stokes equations. Zapiski POMI 397, 150–156 (2011)
Seregin, G.: Remark on Wolf’s condition for boundary regularity of Navier–Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 444, Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 45, 124–132 (2016)
Seregin, G.A., Shilkin, T.N.: Liouville-type theorems for the Navier–Stokes equations. Russ. Math. Surv. 73(4), 661–724 (2018)
Seregin, G., Sverak, V.: On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations. Commun. PDE’s 34, 171–201 (2009)
Seregin, G., Zajaczkowski, W.: A sufficient condition of regularity for axially symmetric solutions to the Navier–Stokes equations. SIAM J. Math. Anal. 39, 669–685 (2007)
Seregin, G., Zhou, D.: Regularity of solutions to the Navier–Stokes equations in \(\dot{B}^{-1}_{\infty, \infty }\). Zapiski POMI 407, 119–128 (2018)
Solonnikov, V.A.: Estimates of solutions to the non-stationary Navier–Stokes system. Zapiski Nauchn. Seminar LOMI 28, 153–231 (1973)
Solonnikov, V.A.: Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator. (Russian) Uspekhi Mat. Nauk 58 (2003), no. 2(350), 123–156; translation in Russian Math. Surveys 58, no. 2, 331–365 (2003)
Ukhovskij, M.R., Yudovich, V.L.: Axially symmetric motions of ideal and viscous fluids filling all space. Prikl. Mat. Mech. 32, 59–69 (1968)
Zhang, P., Zhang, T.: Global axisymmetric solutions to the three-dimensional Navier–Stokes equations system. Int. Math. Res. Not. 2014, 610–642 (2014)
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The work is supported by the Grant RFBR 20-01-00397.
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Dedicated to Vladimir Gilelevich Mazya.
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Seregin, G. Local regularity of axisymmetric solutions to the Navier–Stokes equations. Anal.Math.Phys. 10, 46 (2020). https://doi.org/10.1007/s13324-020-00392-1
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DOI: https://doi.org/10.1007/s13324-020-00392-1